Genetic algorithms in coding theory — a table for A3(n,d)

Abstract

We consider the problem of finding values of A3(n, d), i.e. the maximal size of a ternary code of length n and minimum distance d. Our approach is based on a search for good lower bounds and a comparison of these bounds with known upper bounds. Several lower bounds are obtained using a genetic local search algorithm. Other lower bounds are obtained by constructing codes. For those cases in which lower and upper bounds coincide, this yields exact values of A3(n, d). A table is included containing the known values of the upper and lower bounds for A3(n, d), with n ::; 16. For some values of nand d the corresponding codes are given.

A Connection Between Coding Theory and Polarized Partition Relations

“…Others results and a table of lower and upper bounds for A 2 (n, d) appears in [18], while [21] presents a similar table for A 3 (n, d), with n ≤ 16.…”

“…Consequently, the bound of Theorem 7 is tight at least in this case. Moreover, by using (4), Theorem 7 and tables in [18,21], some new Zarankiewicz numbers can be completely determined, as for example: Z 2,6 (12, 22) = 133, Z 2,7 (8, 28) = 113, Z 2,8 (16, 30) = 241. Table 1 below summarizes applications of the present method for some values of A q (n, d), when q = 2 or 3.…”

“…Table 1 below summarizes applications of the present method for some values of A q (n, d), when q = 2 or 3. The numbers A q (n, d) (first column) are presented in [18,21]. For these numbers, the corresponding polarized partitions (second column) and lower bounds for Zarankiewicz function (third column) are obtained by Theorems 1 and 7, respectively.…”

“…Indeed, the topic is so short of construction that the only exact value known is P 3 (2, 2) = 11, by Exoo [8]. However, making use of the table for A 3 (n, d) due to Vaessen et al [21], we derive good bounds for P 3 (2, t) when t is small.…”

Bounds on mixed binary/ternary codes

“…Another table was given in Vaessens, Aarts, and van Lint [42]. We know of 19 improvements to the latter table, and give an updated version in Table I In [34] it was shown that (using a variation on Proposition 5.8).…”